Second Chern crystals with inherently non-trivial topology.

Chern insulators have been generalized to many classical wave systems and thereby lead to many potential applications such as robust waveguides, quantum computation and high-performance lasers. However, the band structure of a material can be either topologically trivial or non-trivial, depending on how the crystal structure is designed. Here, we propose a second Chern crystal in a four-dimensional parameter space by introducing two extra synthetic translation dimensions. Since the topology of the bulk bands in the synthetic translation space is intrinsically non-trivial, our proposed four-dimensional crystal is guaranteed to be topologically non-trivial regardless of the crystal's detailed configuration. We derive the topologically protected modes on the lower dimensional boundaries of such a crystal via dimension reduction. Remarkably, we observe the one-dimensional gapless dislocation modes and confirm their robustness in experiments. Our findings provide novel perspectives on topologically non-trivial crystals and may inspire designs of classical wave devices.